# Optimal Energy Beamforming under Per-Antenna Power Constraint

## TL;DR

This paper investigates the optimal design of energy beamforming for wireless power transfer under practical per-antenna power constraints, providing optimal, closed-form, and near-optimal solutions.

## Contribution

It introduces the optimal beamforming structure under per-antenna power constraints and derives closed-form solutions for special cases, advancing practical wireless energy transfer methods.

## Key findings

- Sending a single energy beam is optimal under PAC.
- Optimal beamforming has a rank-one structure.
- Sub-optimal solutions perform close to the optimal.

## Abstract

Energy beamforming (EB) is a key technique to enhance the efficiency of wireless power transfer (WPT). In this paper, we study the optimal EB under per-antenna power constraint (PAC) which is more practical than the conventional sum-power constraint (SPC). We consider a multi antenna energy transmitter (ET) with PAC that broadcasts wireless energy to multiple randomly placed energy receivers (ER)s within its cell area. We consider sum energy maximization problem with PAC and provide the optimal solution structure for the general case. This optimal structure implies that sending one energy beam is optimal under PAC which means that the rank of transmit covariance matrix is one similar to SPC. We also derive closed-form solutions for two special cases and propose two sub-optimal solutions for general case, which performs very close to optimal beamforming.